Uniform moderate deviations of functional empirical processes of Markov chains (Q2772051)

Let \(\{X_j,\;j\geq 0\}\) be an ergodic Markov chain with values in a general measurable state space with transition probability \(P\) and with invariant probability measure \(\pi\). Put \(M_n(t)=b^{-1}_n\sum^{[nt]}_{j=0} (\delta_{X_j}-\pi)\) and \((M_n(t))(f)=b^{-1}_n\sum^{[nt]}_{k=0} f(X_k)\) for \(f\) bounded measurable such that \(\pi(f)=0\). The paper obtains the asymptotic estimation of \({\mathbf P}_\mu\{(M_n(\cdot)),(f)\in\cdot\}\) and relative results when \(b_n\) satisfies \(b_n/\sqrt{n}\to+\infty\), \(b_n/n\to 0\).